Ciro Santilli @cirosantilli 37

Postulates are what we now call axioms.

There are 5: en.wikipedia.org/w/index.php?title=Euclidean_geometry&oldid=1036511366#Axioms, the parallel postulate being the most controversial/interesting.

Eugene's background: www.quora.com/Who-is-Eugene-Khutoryansky/answer/Ciro-Santilli

pubmed.ncbi.nlm.nih.gov/27185558/ A Eukaryote without a Mitochondrial Organelle by Karnkowska et. al (2016)

Can be denoted either by:Our default symbol is going to be:

The exact same problem appears over and over, e.g.:

Ciro Santilli also identified knowledge version of this problem: the missing link between basic and advanced.

That's Ciro Santilli's favorite. Of course, there is a huge difference between physical and non physical jobs. But one could start with replacing desk jobs!

Let's start with the one dimensional case. Let the $q:\mathbb{R}\to \mathbb{R}$ and a Functional $I$ defined by a function of three variables $F:{\mathbb{R}}^3\to \mathbb{R}$:

Then, the Euler-Lagrange equation gives the maxima and minima of the that type of functional. Note that this type of functional is just one very specific type of functional amongst all possible functionals that one might come up with. However, it turns out to be enough to do most of physics, so we are happy with with it.

Given $F$, the Euler-Lagrange equations are a system of ordinary differential equations constructed from that $F$ such that the solutions to that system are the maxima/minima.

In the one dimensional case, the system has a single ordinary differential equation:

By $\frac{\partial F}{\partial q}$ and $\frac{\partial F}{\partial \dot{q}}$ we simply mean "the partial derivative of $F$ with respect to its second and third arguments". The notation is a bit confusing at first, but that's all it means.

Therefore, that expression ends up being at most a second order ordinary differential equation where $q$ is the unknown, since:

Now let's think about the multi-dimensional case. Instead of having $q:\mathbb{R}\to \mathbb{R}$, we now have $q:\mathbb{R}\to {\mathbb{R}}^n$. Think about the Lagrangian mechanics motivation of a double pendulum where for a given time we have two angles.

Let's do the 2-dimensional case then. In that case, $F$ is going to be a function of 5 variables $F:{\mathbb{R}}^5\to \mathbb{R}$ rather than 3 as in the one dimensional case, and the functional looks like:

This time, the Euler-Lagrange equations are going to be a system of two ordinary differential equations on two unknown functions $q_1$ and $q_2$ of order up to 2 in both variables:At this point, notation is getting a bit clunky, so people will often condense the $q_i$ vectoror just omit the arguments of $F$ entirely:

Europe has made good regulations to limit the absolute power of immoral companies. Partly because it does not have any companies anymore, but so be it.

But the law that forces every fucking website to show a message "Do you consent to cookies?" is not one of them.

Ciro cannot stand fucking clicking the "I consent" button anymore.

Please stop, for the love of God.

At most, there must be a standardized API that allows your browser to say "I agree or I disagree" automatically to all of them, e.g. an HTTP header.

2021: United Kingdom is considering it post-Brexit: techcrunch.com/2021/09/06/after-years-of-inaction-against-adtech-uks-ico-calls-for-browser-level-controls-to-fix-cookie-fatigue/. Something good might actually come out of Brexit!

Sample usage by Andy Matuschak (possible coiner): notes.andymatuschak.org/About_these_notes